Question 6
The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are now restacked into a single stack by taking cards alternately from the tops of pile $B$ and pile $A,$ respectively. In this process, card number $(n + 1)$ is the bottom card of the new stack, card number $1$ is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is called *magical*. For example, eight cards form a magical stack because cards number $3$ and number $6$ retain their original positions. Find the number of cards in the magical stack in which card number $131$ retains its original position.